A coordinate system, which uses the distance from the coordinate system center and positional angles to identify points in space rather than orthogonal independent unit vectors as in the Cartesian coordinate system. Polar and partly polar (cylindrical) coordinate systems are widely used to describe spin motion in NMR experiments.
It is important to know how to compute the coordinates of a point in the polar coordinate system when they are given in a Cartesian system and vice versa. The length of the vector r pointing from the coordinate origin to a point in 2D space is given as
r = √(x² + y²).
while the polar or phase angle f is obtained by performing the operation
f = arctan (y/x),
where the arctan function is the inverse of the tangent function.

(GRE - sequence) A gradient echo is generated by using a pair of bipolar gradient pulses. In the pulse sequence timing diagram, the basic gradient echo sequence is illustrated. There is no refocusing 180° pulse and the data are sampled during a gradient echo, which is achieved by dephasing the spins with a negatively pulsed gradient before they are rephased by an opposite gradient with opposite polarity to generate the echo.
See also the Pulse Sequence Timing Diagram. There you will find a description of the components.
The excitation pulse is termed the alpha pulse α. It tilts the magnetization by a flip angle α, which is typically between 0° and 90°. With a small flip angle there is a reduction in the value of transverse magnetization that will affect subsequent RF pulses. The flip angle can also be slowly increased during data acquisition (variable flip angle: tilt optimized nonsaturation excitation). The data are not acquired in a steady state, where z-magnetization recovery and destruction by ad-pulses are balanced. However, the z-magnetization is used up by tilting a little more of the remaining z-magnetization into the xy-plane for each acquired imaging line.
Gradient echo imaging is typically accomplished by examining the FID, whereas the read gradient is turned on for localization of the signal in the readout direction. T2* is the characteristic decay time constant associated with the FID. The contrast and signal generated by a gradient echo depend on the size of the longitudinal magnetization and the flip angle. When α = 90° the sequence is identical to the so-called partial saturation or saturation recovery pulse sequence. In standard GRE imaging, this basic pulse sequence is repeated as many times as image lines have to be acquired. Additional gradients or radio frequency pulses are introduced with the aim to spoil to refocus the xy-magnetization at the moment when the spin system is subject to the next α pulse.
As a result of the short repetition time, the z-magnetization cannot fully recover and after a few initial α pulses there is an equilibrium established between z-magnetization recovery and z-magnetization reduction due to the α pulses.
Gradient echoes have a lower SAR, are more sensitive to field inhomogeneities and have a reduced crosstalk, so that a small or no slice gap can be used. In or out of phase imaging depending on the selected TE (and field strength of the magnet) is possible. As the flip angle is decreased, T1 weighting can be maintained by reducing the TR. T2* weighting can be minimized by keeping the TE as short as possible, but pure T2 weighting is not possible. By using a reduced flip angle, some of the magnetization value remains longitudinal (less time needed to achieve full recovery) and for a certain T1 and TR, there exist one flip angle that will give the most signal, known as the "Ernst angle".
Contrast values:
PD weighted: Small flip angle (no T1), long TR (no T1) and short TE (no T2*)
T1 weighted: Large flip angle (70°), short TR (less than 50ms) and short TE
T2* weighted: Small flip angle, some longer TR (100 ms) and long TE (20 ms)

Classification of GRE sequences can be made into four categories:

• T1 weighted or incoherent/(RF or gradient) spoiled GRE sequences
• T1/T2* weighted or coherent/refocused GRE sequences
• T2 weighted contrast enhanced GRE sequences
• ultrafast GRE sequences

See also Gradient Recalled Echo Sequence, Spoiled Gradient Echo Sequence, Refocused Gradient Echo Sequence, Ultrafast Gradient Echo Sequence.

(FLASH) A fast sequence producing signals called gradient echo with low flip angles. FLASH sequences are modifications, which incorporate or remove the effects of transverse coherence respectively.
FLASH uses a semi-random spoiler gradient after each echo to spoil the steady state (to destroy any remaining transverse magnetization) by causing a spatially dependent phase shift. The transverse steady state is spoiled but the longitudinal steady state depends on the T1 values and the flip angle. Extremely short TR times are possible, as a result the sequence provides a mechanism for gaining extremely high T1 contrast by imaging with TR times as brief as 20 to 30 msec while retaining reasonable signal levels. It is important to keep the TE as short as possible to suppress susceptibility artifacts.
The T1 contrast depends on the TR as well as on flip angle, with short TE.
Small flip angles and short TR results in proton density, and long TR in T2* weighting.
With large flip angles and short TR result T1 weighted images.

TR and flip angle adjustment:
TR 3000 ms, Flip Angle 90°
TR 1500 ms, Flip Angle 45°
TR 700 ms, Flip Angle 25°
TR 125 ms, Flip Angle 10°

The apparent ability to trade TR against flip angle for purposes of contrast and the variation in SNR as the scan time (TR) is reduced.

See also Gradient Echo Sequence.

(GEMS) This pulse sequence uses a changeable flip angle instead of a 90° pulse and a gradient instead of a RF pulse to rephase the FID.
T2*, T1 weighted and proton density images can be acquired. The flip angle in combination with the TR determines the T1 weighting and the TE controls the amount of dephasing. To minimize T2* the echo time should be short.

See also Gradient Echo Sequence.